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Old 2018-08-20, 19:45   #1
Just call me Henry
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Sep 2007
Liverpool (GMT/BST)

135468 Posts
Default Concatenated Triangular Numbers

Recently someone mentioned to me at a meal that their favourite number was 136 because it was the first 3 triangular numbers concatenated together and was also triangular itself. I wondered whether there were any more numbers with this property.

I ran a search today and found no more sets of 3 consecutive triangular numbers that are triangular.

Is there any particular reason why this should be the case other than I just haven't searched far enough?

I suppose there is pretty good probabilistic argument for there to be no more. The probability of a random number x being triangular is \(O(1/{\sqrt{2x}})\). The size of the number that needs to be triangular is O(n^6) for the nth set of 3 consecutive triangular numbers concatenated together. I believe that the sum of the probabilities probably converges to a little above 1.

Is anyone able to beat my probabilistic argument for there to be no more?
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